Here we explain how to describe some very important

elements of the absolute Galois group G. For each prime

number p there is a set of elements of G called Fro-

benius elements at p. As usual in mathematics, they are

important partly because we can describe them with some

explicitness, and partly because we can do something

interesting with them. What we will do with them in this

book is to make generalized reciprocity laws, beginning in

the next chapter.

Frobenius elements are difﬁcult to deﬁne precisely. We

give an explanation in this chapter of a working deﬁnition.

The completely correct characterization is presented in a

(probably opaque) appendix to this chapter, so you can

see what it is like. We also discuss “ramiﬁcation” in

this chapter; another appendix gives added details. These

appendices brieﬂy deﬁne many concepts whose detailed

explication would form a large chunk of a text on algebraic

number theory.

Something for Nothing

Ferdinand Georg Frobenius was the nineteenth-century German

mathematician who invented the method of using characters to

178 CHAPTER 16

study group representations.

1

Today, his name is used for (among

other things) particular elements in the absolute Galois group G,

and also their restriction to the Galois groups of various polyno-

mials, that is, for particular permutations of the roots of various

Z-polynomials.

Whenever you can get something for nothing in mathematics,

you take it. Of course, “nothing” is a relative term. What we

mean in this case is that we can deﬁne certain elements of every

Galois group by using an easily applied general theorem that tells

us signiﬁcant information about them. These are the Frobenius

elements.

To take a simpler example ﬁrst, let L be the ﬁeld Q(f )forsome

Z-polynomial f . We know that L is composed of certain complex

numbers. Let τ be complex conjugation.

2

Then τ deﬁnes an element

of the Galois group G(f )ofL:Ifx + iy is any number in L, τ (x + iy)

is again a number in L,andτ respects addition, subtraction,

multiplication, and division. Also, we have a formula for it: τ (x +

iy) = x − iy. Thus, τ is always there, and it is an important element

of the Galois group G(f ). It is the neutral element if and only if

L actually contains only real numbers, because then L does not

contain any numbers of the form x + iy with y = 0. W e can make τ

into an honorary Frobenius element at inﬁnity, and call it Frob

∞

.

This is purely a notational convention, and does not help us to

deﬁne Frob

p

when p is a prime. However, this notation Frob

∞

is

common in the research literature.

The other Frobenius elements are at p for each prime p,and

we now begin to deﬁne them. If you choose to skip the rest of this

discussion, you need to know only:

•

Frob

p

is a particular element of G.

•

Actually, this is a lie, because we cannot deﬁne Frob

p

so

precisely. Really, given p, there is deﬁned a certain

conjugacy class in G (depending on p), and Frob

p

is taken

1

In fact, Frobenius deserves credit for inventing group representations themselves.

2

We have used the letter τ before to stand for various things, but here we will use

it for the speciﬁc function of complex conjugation. We have been using c for complex

conjugation, but from now on we prefer Greek letters for elements of Galois groups.

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